# Properties

 Label 9680.2.a.ct.1.1 Level $9680$ Weight $2$ Character 9680.1 Self dual yes Analytic conductor $77.295$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9680 = 2^{4} \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9680.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$77.2951891566$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.725.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 3x^{2} + x + 1$$ x^4 - x^3 - 3*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$0.737640$$ of defining polynomial Character $$\chi$$ $$=$$ 9680.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.19353 q^{3} +1.00000 q^{5} -1.83785 q^{7} -1.57549 q^{9} +O(q^{10})$$ $$q-1.19353 q^{3} +1.00000 q^{5} -1.83785 q^{7} -1.57549 q^{9} +1.25546 q^{13} -1.19353 q^{15} -6.44899 q^{17} +1.25120 q^{19} +2.19353 q^{21} +4.35567 q^{23} +1.00000 q^{25} +5.46097 q^{27} +0.726479 q^{29} +11.0790 q^{31} -1.83785 q^{35} -11.5086 q^{37} -1.49843 q^{39} -7.92427 q^{41} +0.606873 q^{43} -1.57549 q^{45} +3.98194 q^{47} -3.62230 q^{49} +7.69704 q^{51} +6.83290 q^{53} -1.49334 q^{57} +1.41201 q^{59} +9.75208 q^{61} +2.89552 q^{63} +1.25546 q^{65} -12.5899 q^{67} -5.19862 q^{69} +6.79665 q^{71} +10.0652 q^{73} -1.19353 q^{75} +6.51977 q^{79} -1.79134 q^{81} +0.644326 q^{83} -6.44899 q^{85} -0.867073 q^{87} +9.39723 q^{89} -2.30735 q^{91} -13.2231 q^{93} +1.25120 q^{95} -4.66191 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 4 * q^5 - 7 * q^7 - 4 * q^9 $$4 q + 2 q^{3} + 4 q^{5} - 7 q^{7} - 4 q^{9} - 3 q^{13} + 2 q^{15} - 11 q^{17} + 2 q^{19} + 2 q^{21} + 11 q^{23} + 4 q^{25} - q^{27} - 4 q^{29} + 17 q^{31} - 7 q^{35} - 3 q^{37} + q^{39} - 13 q^{41} - 7 q^{43} - 4 q^{45} + q^{47} - 5 q^{49} - q^{51} - 15 q^{53} - 17 q^{57} + 17 q^{59} - 4 q^{61} + 15 q^{63} - 3 q^{65} - 7 q^{67} + 4 q^{69} + 15 q^{71} - 7 q^{73} + 2 q^{75} - 12 q^{79} - 8 q^{81} + 9 q^{83} - 11 q^{85} - 23 q^{87} - 12 q^{89} + 24 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97}+O(q^{100})$$ 4 * q + 2 * q^3 + 4 * q^5 - 7 * q^7 - 4 * q^9 - 3 * q^13 + 2 * q^15 - 11 * q^17 + 2 * q^19 + 2 * q^21 + 11 * q^23 + 4 * q^25 - q^27 - 4 * q^29 + 17 * q^31 - 7 * q^35 - 3 * q^37 + q^39 - 13 * q^41 - 7 * q^43 - 4 * q^45 + q^47 - 5 * q^49 - q^51 - 15 * q^53 - 17 * q^57 + 17 * q^59 - 4 * q^61 + 15 * q^63 - 3 * q^65 - 7 * q^67 + 4 * q^69 + 15 * q^71 - 7 * q^73 + 2 * q^75 - 12 * q^79 - 8 * q^81 + 9 * q^83 - 11 * q^85 - 23 * q^87 - 12 * q^89 + 24 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −1.19353 −0.689083 −0.344542 0.938771i $$-0.611966\pi$$
−0.344542 + 0.938771i $$0.611966\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −1.83785 −0.694643 −0.347322 0.937746i $$-0.612909\pi$$
−0.347322 + 0.937746i $$0.612909\pi$$
$$8$$ 0 0
$$9$$ −1.57549 −0.525164
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ 1.25546 0.348202 0.174101 0.984728i $$-0.444298\pi$$
0.174101 + 0.984728i $$0.444298\pi$$
$$14$$ 0 0
$$15$$ −1.19353 −0.308167
$$16$$ 0 0
$$17$$ −6.44899 −1.56411 −0.782055 0.623210i $$-0.785828\pi$$
−0.782055 + 0.623210i $$0.785828\pi$$
$$18$$ 0 0
$$19$$ 1.25120 0.287045 0.143522 0.989647i $$-0.454157\pi$$
0.143522 + 0.989647i $$0.454157\pi$$
$$20$$ 0 0
$$21$$ 2.19353 0.478667
$$22$$ 0 0
$$23$$ 4.35567 0.908221 0.454110 0.890945i $$-0.349957\pi$$
0.454110 + 0.890945i $$0.349957\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ 5.46097 1.05097
$$28$$ 0 0
$$29$$ 0.726479 0.134904 0.0674519 0.997723i $$-0.478513\pi$$
0.0674519 + 0.997723i $$0.478513\pi$$
$$30$$ 0 0
$$31$$ 11.0790 1.98985 0.994924 0.100626i $$-0.0320844\pi$$
0.994924 + 0.100626i $$0.0320844\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −1.83785 −0.310654
$$36$$ 0 0
$$37$$ −11.5086 −1.89200 −0.946001 0.324162i $$-0.894918\pi$$
−0.946001 + 0.324162i $$0.894918\pi$$
$$38$$ 0 0
$$39$$ −1.49843 −0.239940
$$40$$ 0 0
$$41$$ −7.92427 −1.23756 −0.618781 0.785563i $$-0.712373\pi$$
−0.618781 + 0.785563i $$0.712373\pi$$
$$42$$ 0 0
$$43$$ 0.606873 0.0925473 0.0462736 0.998929i $$-0.485265\pi$$
0.0462736 + 0.998929i $$0.485265\pi$$
$$44$$ 0 0
$$45$$ −1.57549 −0.234861
$$46$$ 0 0
$$47$$ 3.98194 0.580826 0.290413 0.956901i $$-0.406207\pi$$
0.290413 + 0.956901i $$0.406207\pi$$
$$48$$ 0 0
$$49$$ −3.62230 −0.517471
$$50$$ 0 0
$$51$$ 7.69704 1.07780
$$52$$ 0 0
$$53$$ 6.83290 0.938571 0.469285 0.883047i $$-0.344512\pi$$
0.469285 + 0.883047i $$0.344512\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −1.49334 −0.197798
$$58$$ 0 0
$$59$$ 1.41201 0.183828 0.0919141 0.995767i $$-0.470701\pi$$
0.0919141 + 0.995767i $$0.470701\pi$$
$$60$$ 0 0
$$61$$ 9.75208 1.24863 0.624313 0.781174i $$-0.285379\pi$$
0.624313 + 0.781174i $$0.285379\pi$$
$$62$$ 0 0
$$63$$ 2.89552 0.364802
$$64$$ 0 0
$$65$$ 1.25546 0.155721
$$66$$ 0 0
$$67$$ −12.5899 −1.53811 −0.769053 0.639186i $$-0.779272\pi$$
−0.769053 + 0.639186i $$0.779272\pi$$
$$68$$ 0 0
$$69$$ −5.19862 −0.625840
$$70$$ 0 0
$$71$$ 6.79665 0.806614 0.403307 0.915065i $$-0.367861\pi$$
0.403307 + 0.915065i $$0.367861\pi$$
$$72$$ 0 0
$$73$$ 10.0652 1.17804 0.589022 0.808117i $$-0.299513\pi$$
0.589022 + 0.808117i $$0.299513\pi$$
$$74$$ 0 0
$$75$$ −1.19353 −0.137817
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 6.51977 0.733531 0.366765 0.930313i $$-0.380465\pi$$
0.366765 + 0.930313i $$0.380465\pi$$
$$80$$ 0 0
$$81$$ −1.79134 −0.199038
$$82$$ 0 0
$$83$$ 0.644326 0.0707239 0.0353620 0.999375i $$-0.488742\pi$$
0.0353620 + 0.999375i $$0.488742\pi$$
$$84$$ 0 0
$$85$$ −6.44899 −0.699491
$$86$$ 0 0
$$87$$ −0.867073 −0.0929600
$$88$$ 0 0
$$89$$ 9.39723 0.996104 0.498052 0.867147i $$-0.334049\pi$$
0.498052 + 0.867147i $$0.334049\pi$$
$$90$$ 0 0
$$91$$ −2.30735 −0.241876
$$92$$ 0 0
$$93$$ −13.2231 −1.37117
$$94$$ 0 0
$$95$$ 1.25120 0.128370
$$96$$ 0 0
$$97$$ −4.66191 −0.473345 −0.236673 0.971589i $$-0.576057\pi$$
−0.236673 + 0.971589i $$0.576057\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −10.1078 −1.00576 −0.502880 0.864356i $$-0.667726\pi$$
−0.502880 + 0.864356i $$0.667726\pi$$
$$102$$ 0 0
$$103$$ 3.71135 0.365690 0.182845 0.983142i $$-0.441469\pi$$
0.182845 + 0.983142i $$0.441469\pi$$
$$104$$ 0 0
$$105$$ 2.19353 0.214066
$$106$$ 0 0
$$107$$ −7.69278 −0.743689 −0.371845 0.928295i $$-0.621275\pi$$
−0.371845 + 0.928295i $$0.621275\pi$$
$$108$$ 0 0
$$109$$ 3.06569 0.293640 0.146820 0.989163i $$-0.453096\pi$$
0.146820 + 0.989163i $$0.453096\pi$$
$$110$$ 0 0
$$111$$ 13.7358 1.30375
$$112$$ 0 0
$$113$$ −10.5168 −0.989341 −0.494670 0.869081i $$-0.664711\pi$$
−0.494670 + 0.869081i $$0.664711\pi$$
$$114$$ 0 0
$$115$$ 4.35567 0.406169
$$116$$ 0 0
$$117$$ −1.97797 −0.182864
$$118$$ 0 0
$$119$$ 11.8523 1.08650
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 9.45783 0.852784
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −5.97978 −0.530620 −0.265310 0.964163i $$-0.585474\pi$$
−0.265310 + 0.964163i $$0.585474\pi$$
$$128$$ 0 0
$$129$$ −0.724319 −0.0637728
$$130$$ 0 0
$$131$$ −9.13159 −0.797831 −0.398915 0.916988i $$-0.630613\pi$$
−0.398915 + 0.916988i $$0.630613\pi$$
$$132$$ 0 0
$$133$$ −2.29952 −0.199394
$$134$$ 0 0
$$135$$ 5.46097 0.470006
$$136$$ 0 0
$$137$$ 5.48597 0.468698 0.234349 0.972153i $$-0.424704\pi$$
0.234349 + 0.972153i $$0.424704\pi$$
$$138$$ 0 0
$$139$$ −17.7510 −1.50562 −0.752808 0.658240i $$-0.771301\pi$$
−0.752808 + 0.658240i $$0.771301\pi$$
$$140$$ 0 0
$$141$$ −4.75255 −0.400237
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0.726479 0.0603308
$$146$$ 0 0
$$147$$ 4.32331 0.356581
$$148$$ 0 0
$$149$$ 15.4005 1.26166 0.630829 0.775922i $$-0.282715\pi$$
0.630829 + 0.775922i $$0.282715\pi$$
$$150$$ 0 0
$$151$$ −14.4892 −1.17912 −0.589558 0.807726i $$-0.700698\pi$$
−0.589558 + 0.807726i $$0.700698\pi$$
$$152$$ 0 0
$$153$$ 10.1603 0.821415
$$154$$ 0 0
$$155$$ 11.0790 0.889887
$$156$$ 0 0
$$157$$ −4.88487 −0.389855 −0.194928 0.980818i $$-0.562447\pi$$
−0.194928 + 0.980818i $$0.562447\pi$$
$$158$$ 0 0
$$159$$ −8.15525 −0.646753
$$160$$ 0 0
$$161$$ −8.00509 −0.630889
$$162$$ 0 0
$$163$$ 23.0578 1.80603 0.903013 0.429612i $$-0.141350\pi$$
0.903013 + 0.429612i $$0.141350\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3.24094 0.250791 0.125396 0.992107i $$-0.459980\pi$$
0.125396 + 0.992107i $$0.459980\pi$$
$$168$$ 0 0
$$169$$ −11.4238 −0.878755
$$170$$ 0 0
$$171$$ −1.97125 −0.150746
$$172$$ 0 0
$$173$$ −17.6970 −1.34548 −0.672741 0.739878i $$-0.734883\pi$$
−0.672741 + 0.739878i $$0.734883\pi$$
$$174$$ 0 0
$$175$$ −1.83785 −0.138929
$$176$$ 0 0
$$177$$ −1.68527 −0.126673
$$178$$ 0 0
$$179$$ −17.6078 −1.31607 −0.658036 0.752987i $$-0.728612\pi$$
−0.658036 + 0.752987i $$0.728612\pi$$
$$180$$ 0 0
$$181$$ −4.79712 −0.356567 −0.178284 0.983979i $$-0.557054\pi$$
−0.178284 + 0.983979i $$0.557054\pi$$
$$182$$ 0 0
$$183$$ −11.6394 −0.860407
$$184$$ 0 0
$$185$$ −11.5086 −0.846129
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −10.0365 −0.730046
$$190$$ 0 0
$$191$$ 11.6292 0.841459 0.420730 0.907186i $$-0.361774\pi$$
0.420730 + 0.907186i $$0.361774\pi$$
$$192$$ 0 0
$$193$$ −4.05323 −0.291758 −0.145879 0.989302i $$-0.546601\pi$$
−0.145879 + 0.989302i $$0.546601\pi$$
$$194$$ 0 0
$$195$$ −1.49843 −0.107305
$$196$$ 0 0
$$197$$ 17.4066 1.24017 0.620084 0.784536i $$-0.287099\pi$$
0.620084 + 0.784536i $$0.287099\pi$$
$$198$$ 0 0
$$199$$ −22.0409 −1.56244 −0.781218 0.624259i $$-0.785401\pi$$
−0.781218 + 0.624259i $$0.785401\pi$$
$$200$$ 0 0
$$201$$ 15.0264 1.05988
$$202$$ 0 0
$$203$$ −1.33516 −0.0937100
$$204$$ 0 0
$$205$$ −7.92427 −0.553455
$$206$$ 0 0
$$207$$ −6.86233 −0.476965
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −8.27193 −0.569463 −0.284731 0.958607i $$-0.591904\pi$$
−0.284731 + 0.958607i $$0.591904\pi$$
$$212$$ 0 0
$$213$$ −8.11198 −0.555824
$$214$$ 0 0
$$215$$ 0.606873 0.0413884
$$216$$ 0 0
$$217$$ −20.3616 −1.38223
$$218$$ 0 0
$$219$$ −12.0131 −0.811770
$$220$$ 0 0
$$221$$ −8.09646 −0.544627
$$222$$ 0 0
$$223$$ −10.8470 −0.726372 −0.363186 0.931717i $$-0.618311\pi$$
−0.363186 + 0.931717i $$0.618311\pi$$
$$224$$ 0 0
$$225$$ −1.57549 −0.105033
$$226$$ 0 0
$$227$$ −5.43321 −0.360615 −0.180308 0.983610i $$-0.557709\pi$$
−0.180308 + 0.983610i $$0.557709\pi$$
$$228$$ 0 0
$$229$$ 20.8976 1.38095 0.690476 0.723355i $$-0.257401\pi$$
0.690476 + 0.723355i $$0.257401\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −26.9714 −1.76696 −0.883478 0.468472i $$-0.844805\pi$$
−0.883478 + 0.468472i $$0.844805\pi$$
$$234$$ 0 0
$$235$$ 3.98194 0.259753
$$236$$ 0 0
$$237$$ −7.78152 −0.505464
$$238$$ 0 0
$$239$$ 23.2655 1.50492 0.752459 0.658639i $$-0.228867\pi$$
0.752459 + 0.658639i $$0.228867\pi$$
$$240$$ 0 0
$$241$$ −9.93604 −0.640037 −0.320018 0.947411i $$-0.603689\pi$$
−0.320018 + 0.947411i $$0.603689\pi$$
$$242$$ 0 0
$$243$$ −14.2449 −0.913811
$$244$$ 0 0
$$245$$ −3.62230 −0.231420
$$246$$ 0 0
$$247$$ 1.57083 0.0999497
$$248$$ 0 0
$$249$$ −0.769020 −0.0487347
$$250$$ 0 0
$$251$$ 17.8421 1.12618 0.563092 0.826394i $$-0.309612\pi$$
0.563092 + 0.826394i $$0.309612\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 7.69704 0.482008
$$256$$ 0 0
$$257$$ −27.9332 −1.74243 −0.871214 0.490903i $$-0.836667\pi$$
−0.871214 + 0.490903i $$0.836667\pi$$
$$258$$ 0 0
$$259$$ 21.1511 1.31427
$$260$$ 0 0
$$261$$ −1.14456 −0.0708467
$$262$$ 0 0
$$263$$ −14.8691 −0.916871 −0.458436 0.888728i $$-0.651590\pi$$
−0.458436 + 0.888728i $$0.651590\pi$$
$$264$$ 0 0
$$265$$ 6.83290 0.419742
$$266$$ 0 0
$$267$$ −11.2158 −0.686399
$$268$$ 0 0
$$269$$ −7.98061 −0.486586 −0.243293 0.969953i $$-0.578228\pi$$
−0.243293 + 0.969953i $$0.578228\pi$$
$$270$$ 0 0
$$271$$ −18.8092 −1.14258 −0.571290 0.820748i $$-0.693557\pi$$
−0.571290 + 0.820748i $$0.693557\pi$$
$$272$$ 0 0
$$273$$ 2.75389 0.166673
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.8607 0.652554 0.326277 0.945274i $$-0.394206\pi$$
0.326277 + 0.945274i $$0.394206\pi$$
$$278$$ 0 0
$$279$$ −17.4549 −1.04500
$$280$$ 0 0
$$281$$ 24.1095 1.43825 0.719126 0.694880i $$-0.244543\pi$$
0.719126 + 0.694880i $$0.244543\pi$$
$$282$$ 0 0
$$283$$ −9.11974 −0.542112 −0.271056 0.962564i $$-0.587373\pi$$
−0.271056 + 0.962564i $$0.587373\pi$$
$$284$$ 0 0
$$285$$ −1.49334 −0.0884578
$$286$$ 0 0
$$287$$ 14.5636 0.859665
$$288$$ 0 0
$$289$$ 24.5895 1.44644
$$290$$ 0 0
$$291$$ 5.56412 0.326174
$$292$$ 0 0
$$293$$ −26.2288 −1.53230 −0.766151 0.642660i $$-0.777831\pi$$
−0.766151 + 0.642660i $$0.777831\pi$$
$$294$$ 0 0
$$295$$ 1.41201 0.0822105
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 5.46838 0.316245
$$300$$ 0 0
$$301$$ −1.11534 −0.0642873
$$302$$ 0 0
$$303$$ 12.0639 0.693052
$$304$$ 0 0
$$305$$ 9.75208 0.558402
$$306$$ 0 0
$$307$$ −8.16034 −0.465735 −0.232868 0.972508i $$-0.574811\pi$$
−0.232868 + 0.972508i $$0.574811\pi$$
$$308$$ 0 0
$$309$$ −4.42960 −0.251991
$$310$$ 0 0
$$311$$ 13.4933 0.765137 0.382569 0.923927i $$-0.375040\pi$$
0.382569 + 0.923927i $$0.375040\pi$$
$$312$$ 0 0
$$313$$ 7.83820 0.443041 0.221521 0.975156i $$-0.428898\pi$$
0.221521 + 0.975156i $$0.428898\pi$$
$$314$$ 0 0
$$315$$ 2.89552 0.163144
$$316$$ 0 0
$$317$$ 6.32808 0.355421 0.177710 0.984083i $$-0.443131\pi$$
0.177710 + 0.984083i $$0.443131\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 9.18154 0.512464
$$322$$ 0 0
$$323$$ −8.06897 −0.448969
$$324$$ 0 0
$$325$$ 1.25546 0.0696405
$$326$$ 0 0
$$327$$ −3.65898 −0.202342
$$328$$ 0 0
$$329$$ −7.31822 −0.403467
$$330$$ 0 0
$$331$$ 2.92392 0.160713 0.0803566 0.996766i $$-0.474394\pi$$
0.0803566 + 0.996766i $$0.474394\pi$$
$$332$$ 0 0
$$333$$ 18.1317 0.993612
$$334$$ 0 0
$$335$$ −12.5899 −0.687861
$$336$$ 0 0
$$337$$ −22.3306 −1.21642 −0.608211 0.793775i $$-0.708113\pi$$
−0.608211 + 0.793775i $$0.708113\pi$$
$$338$$ 0 0
$$339$$ 12.5521 0.681738
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 19.5222 1.05410
$$344$$ 0 0
$$345$$ −5.19862 −0.279884
$$346$$ 0 0
$$347$$ 5.62309 0.301863 0.150932 0.988544i $$-0.451773\pi$$
0.150932 + 0.988544i $$0.451773\pi$$
$$348$$ 0 0
$$349$$ 4.13639 0.221416 0.110708 0.993853i $$-0.464688\pi$$
0.110708 + 0.993853i $$0.464688\pi$$
$$350$$ 0 0
$$351$$ 6.85605 0.365949
$$352$$ 0 0
$$353$$ 17.3112 0.921380 0.460690 0.887561i $$-0.347602\pi$$
0.460690 + 0.887561i $$0.347602\pi$$
$$354$$ 0 0
$$355$$ 6.79665 0.360729
$$356$$ 0 0
$$357$$ −14.1460 −0.748687
$$358$$ 0 0
$$359$$ 15.0666 0.795187 0.397594 0.917562i $$-0.369845\pi$$
0.397594 + 0.917562i $$0.369845\pi$$
$$360$$ 0 0
$$361$$ −17.4345 −0.917605
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 10.0652 0.526837
$$366$$ 0 0
$$367$$ 5.13586 0.268089 0.134045 0.990975i $$-0.457203\pi$$
0.134045 + 0.990975i $$0.457203\pi$$
$$368$$ 0 0
$$369$$ 12.4846 0.649924
$$370$$ 0 0
$$371$$ −12.5579 −0.651972
$$372$$ 0 0
$$373$$ 4.24462 0.219778 0.109889 0.993944i $$-0.464950\pi$$
0.109889 + 0.993944i $$0.464950\pi$$
$$374$$ 0 0
$$375$$ −1.19353 −0.0616335
$$376$$ 0 0
$$377$$ 0.912067 0.0469738
$$378$$ 0 0
$$379$$ 19.9850 1.02656 0.513280 0.858221i $$-0.328430\pi$$
0.513280 + 0.858221i $$0.328430\pi$$
$$380$$ 0 0
$$381$$ 7.13703 0.365641
$$382$$ 0 0
$$383$$ −7.47296 −0.381850 −0.190925 0.981605i $$-0.561149\pi$$
−0.190925 + 0.981605i $$0.561149\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −0.956124 −0.0486025
$$388$$ 0 0
$$389$$ −23.4196 −1.18742 −0.593709 0.804680i $$-0.702337\pi$$
−0.593709 + 0.804680i $$0.702337\pi$$
$$390$$ 0 0
$$391$$ −28.0897 −1.42056
$$392$$ 0 0
$$393$$ 10.8988 0.549772
$$394$$ 0 0
$$395$$ 6.51977 0.328045
$$396$$ 0 0
$$397$$ 13.2416 0.664578 0.332289 0.943178i $$-0.392179\pi$$
0.332289 + 0.943178i $$0.392179\pi$$
$$398$$ 0 0
$$399$$ 2.74454 0.137399
$$400$$ 0 0
$$401$$ 8.32901 0.415931 0.207965 0.978136i $$-0.433316\pi$$
0.207965 + 0.978136i $$0.433316\pi$$
$$402$$ 0 0
$$403$$ 13.9093 0.692870
$$404$$ 0 0
$$405$$ −1.79134 −0.0890125
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −22.7399 −1.12442 −0.562209 0.826995i $$-0.690048\pi$$
−0.562209 + 0.826995i $$0.690048\pi$$
$$410$$ 0 0
$$411$$ −6.54765 −0.322972
$$412$$ 0 0
$$413$$ −2.59507 −0.127695
$$414$$ 0 0
$$415$$ 0.644326 0.0316287
$$416$$ 0 0
$$417$$ 21.1863 1.03750
$$418$$ 0 0
$$419$$ 15.9446 0.778946 0.389473 0.921038i $$-0.372657\pi$$
0.389473 + 0.921038i $$0.372657\pi$$
$$420$$ 0 0
$$421$$ −33.0997 −1.61318 −0.806589 0.591112i $$-0.798689\pi$$
−0.806589 + 0.591112i $$0.798689\pi$$
$$422$$ 0 0
$$423$$ −6.27352 −0.305029
$$424$$ 0 0
$$425$$ −6.44899 −0.312822
$$426$$ 0 0
$$427$$ −17.9229 −0.867349
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 15.1286 0.728718 0.364359 0.931259i $$-0.381288\pi$$
0.364359 + 0.931259i $$0.381288\pi$$
$$432$$ 0 0
$$433$$ 10.7199 0.515165 0.257583 0.966256i $$-0.417074\pi$$
0.257583 + 0.966256i $$0.417074\pi$$
$$434$$ 0 0
$$435$$ −0.867073 −0.0415730
$$436$$ 0 0
$$437$$ 5.44981 0.260700
$$438$$ 0 0
$$439$$ −35.9820 −1.71733 −0.858663 0.512540i $$-0.828705\pi$$
−0.858663 + 0.512540i $$0.828705\pi$$
$$440$$ 0 0
$$441$$ 5.70690 0.271757
$$442$$ 0 0
$$443$$ −3.65928 −0.173857 −0.0869287 0.996215i $$-0.527705\pi$$
−0.0869287 + 0.996215i $$0.527705\pi$$
$$444$$ 0 0
$$445$$ 9.39723 0.445471
$$446$$ 0 0
$$447$$ −18.3809 −0.869388
$$448$$ 0 0
$$449$$ 7.35426 0.347069 0.173534 0.984828i $$-0.444481\pi$$
0.173534 + 0.984828i $$0.444481\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 17.2933 0.812508
$$454$$ 0 0
$$455$$ −2.30735 −0.108170
$$456$$ 0 0
$$457$$ −31.4949 −1.47327 −0.736635 0.676291i $$-0.763586\pi$$
−0.736635 + 0.676291i $$0.763586\pi$$
$$458$$ 0 0
$$459$$ −35.2178 −1.64382
$$460$$ 0 0
$$461$$ −39.7866 −1.85305 −0.926523 0.376239i $$-0.877217\pi$$
−0.926523 + 0.376239i $$0.877217\pi$$
$$462$$ 0 0
$$463$$ 14.1053 0.655529 0.327764 0.944759i $$-0.393705\pi$$
0.327764 + 0.944759i $$0.393705\pi$$
$$464$$ 0 0
$$465$$ −13.2231 −0.613206
$$466$$ 0 0
$$467$$ −5.15219 −0.238415 −0.119207 0.992869i $$-0.538035\pi$$
−0.119207 + 0.992869i $$0.538035\pi$$
$$468$$ 0 0
$$469$$ 23.1384 1.06843
$$470$$ 0 0
$$471$$ 5.83023 0.268643
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 1.25120 0.0574089
$$476$$ 0 0
$$477$$ −10.7652 −0.492904
$$478$$ 0 0
$$479$$ 16.1277 0.736891 0.368446 0.929649i $$-0.379890\pi$$
0.368446 + 0.929649i $$0.379890\pi$$
$$480$$ 0 0
$$481$$ −14.4486 −0.658800
$$482$$ 0 0
$$483$$ 9.55429 0.434735
$$484$$ 0 0
$$485$$ −4.66191 −0.211686
$$486$$ 0 0
$$487$$ −0.406098 −0.0184020 −0.00920102 0.999958i $$-0.502929\pi$$
−0.00920102 + 0.999958i $$0.502929\pi$$
$$488$$ 0 0
$$489$$ −27.5201 −1.24450
$$490$$ 0 0
$$491$$ 11.8359 0.534147 0.267074 0.963676i $$-0.413943\pi$$
0.267074 + 0.963676i $$0.413943\pi$$
$$492$$ 0 0
$$493$$ −4.68506 −0.211004
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −12.4912 −0.560309
$$498$$ 0 0
$$499$$ −27.3527 −1.22448 −0.612238 0.790674i $$-0.709731\pi$$
−0.612238 + 0.790674i $$0.709731\pi$$
$$500$$ 0 0
$$501$$ −3.86815 −0.172816
$$502$$ 0 0
$$503$$ −18.7444 −0.835771 −0.417885 0.908500i $$-0.637229\pi$$
−0.417885 + 0.908500i $$0.637229\pi$$
$$504$$ 0 0
$$505$$ −10.1078 −0.449789
$$506$$ 0 0
$$507$$ 13.6346 0.605535
$$508$$ 0 0
$$509$$ 35.4484 1.57122 0.785610 0.618722i $$-0.212349\pi$$
0.785610 + 0.618722i $$0.212349\pi$$
$$510$$ 0 0
$$511$$ −18.4984 −0.818320
$$512$$ 0 0
$$513$$ 6.83276 0.301674
$$514$$ 0 0
$$515$$ 3.71135 0.163542
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 21.1219 0.927149
$$520$$ 0 0
$$521$$ −20.4748 −0.897016 −0.448508 0.893779i $$-0.648044\pi$$
−0.448508 + 0.893779i $$0.648044\pi$$
$$522$$ 0 0
$$523$$ −30.2594 −1.32315 −0.661576 0.749878i $$-0.730112\pi$$
−0.661576 + 0.749878i $$0.730112\pi$$
$$524$$ 0 0
$$525$$ 2.19353 0.0957334
$$526$$ 0 0
$$527$$ −71.4484 −3.11234
$$528$$ 0 0
$$529$$ −4.02810 −0.175135
$$530$$ 0 0
$$531$$ −2.22461 −0.0965400
$$532$$ 0 0
$$533$$ −9.94862 −0.430922
$$534$$ 0 0
$$535$$ −7.69278 −0.332588
$$536$$ 0 0
$$537$$ 21.0154 0.906882
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −8.49204 −0.365101 −0.182551 0.983196i $$-0.558435\pi$$
−0.182551 + 0.983196i $$0.558435\pi$$
$$542$$ 0 0
$$543$$ 5.72549 0.245704
$$544$$ 0 0
$$545$$ 3.06569 0.131320
$$546$$ 0 0
$$547$$ 26.0488 1.11377 0.556883 0.830591i $$-0.311997\pi$$
0.556883 + 0.830591i $$0.311997\pi$$
$$548$$ 0 0
$$549$$ −15.3643 −0.655734
$$550$$ 0 0
$$551$$ 0.908970 0.0387234
$$552$$ 0 0
$$553$$ −11.9824 −0.509542
$$554$$ 0 0
$$555$$ 13.7358 0.583054
$$556$$ 0 0
$$557$$ −11.8872 −0.503677 −0.251839 0.967769i $$-0.581035\pi$$
−0.251839 + 0.967769i $$0.581035\pi$$
$$558$$ 0 0
$$559$$ 0.761906 0.0322252
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −38.1919 −1.60959 −0.804797 0.593550i $$-0.797726\pi$$
−0.804797 + 0.593550i $$0.797726\pi$$
$$564$$ 0 0
$$565$$ −10.5168 −0.442447
$$566$$ 0 0
$$567$$ 3.29222 0.138260
$$568$$ 0 0
$$569$$ −29.9184 −1.25425 −0.627123 0.778920i $$-0.715768\pi$$
−0.627123 + 0.778920i $$0.715768\pi$$
$$570$$ 0 0
$$571$$ 34.1271 1.42817 0.714087 0.700057i $$-0.246842\pi$$
0.714087 + 0.700057i $$0.246842\pi$$
$$572$$ 0 0
$$573$$ −13.8798 −0.579835
$$574$$ 0 0
$$575$$ 4.35567 0.181644
$$576$$ 0 0
$$577$$ −5.81087 −0.241910 −0.120955 0.992658i $$-0.538596\pi$$
−0.120955 + 0.992658i $$0.538596\pi$$
$$578$$ 0 0
$$579$$ 4.83764 0.201045
$$580$$ 0 0
$$581$$ −1.18418 −0.0491279
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ −1.97797 −0.0817791
$$586$$ 0 0
$$587$$ −7.05828 −0.291326 −0.145663 0.989334i $$-0.546532\pi$$
−0.145663 + 0.989334i $$0.546532\pi$$
$$588$$ 0 0
$$589$$ 13.8620 0.571175
$$590$$ 0 0
$$591$$ −20.7752 −0.854579
$$592$$ 0 0
$$593$$ −35.1649 −1.44405 −0.722024 0.691868i $$-0.756788\pi$$
−0.722024 + 0.691868i $$0.756788\pi$$
$$594$$ 0 0
$$595$$ 11.8523 0.485897
$$596$$ 0 0
$$597$$ 26.3064 1.07665
$$598$$ 0 0
$$599$$ −45.3634 −1.85350 −0.926750 0.375678i $$-0.877410\pi$$
−0.926750 + 0.375678i $$0.877410\pi$$
$$600$$ 0 0
$$601$$ −42.7574 −1.74411 −0.872055 0.489408i $$-0.837213\pi$$
−0.872055 + 0.489408i $$0.837213\pi$$
$$602$$ 0 0
$$603$$ 19.8354 0.807758
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −39.7376 −1.61290 −0.806450 0.591302i $$-0.798614\pi$$
−0.806450 + 0.591302i $$0.798614\pi$$
$$608$$ 0 0
$$609$$ 1.59355 0.0645740
$$610$$ 0 0
$$611$$ 4.99917 0.202245
$$612$$ 0 0
$$613$$ −47.4880 −1.91802 −0.959011 0.283367i $$-0.908548\pi$$
−0.959011 + 0.283367i $$0.908548\pi$$
$$614$$ 0 0
$$615$$ 9.45783 0.381377
$$616$$ 0 0
$$617$$ 16.6834 0.671649 0.335824 0.941925i $$-0.390985\pi$$
0.335824 + 0.941925i $$0.390985\pi$$
$$618$$ 0 0
$$619$$ 28.9511 1.16364 0.581822 0.813316i $$-0.302340\pi$$
0.581822 + 0.813316i $$0.302340\pi$$
$$620$$ 0 0
$$621$$ 23.7862 0.954508
$$622$$ 0 0
$$623$$ −17.2707 −0.691937
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 74.2189 2.95930
$$630$$ 0 0
$$631$$ −9.97512 −0.397103 −0.198552 0.980090i $$-0.563624\pi$$
−0.198552 + 0.980090i $$0.563624\pi$$
$$632$$ 0 0
$$633$$ 9.87277 0.392407
$$634$$ 0 0
$$635$$ −5.97978 −0.237300
$$636$$ 0 0
$$637$$ −4.54766 −0.180185
$$638$$ 0 0
$$639$$ −10.7081 −0.423605
$$640$$ 0 0
$$641$$ −43.2582 −1.70860 −0.854298 0.519784i $$-0.826012\pi$$
−0.854298 + 0.519784i $$0.826012\pi$$
$$642$$ 0 0
$$643$$ −10.3729 −0.409069 −0.204534 0.978859i $$-0.565568\pi$$
−0.204534 + 0.978859i $$0.565568\pi$$
$$644$$ 0 0
$$645$$ −0.724319 −0.0285200
$$646$$ 0 0
$$647$$ −8.97743 −0.352939 −0.176470 0.984306i $$-0.556468\pi$$
−0.176470 + 0.984306i $$0.556468\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 24.3021 0.952475
$$652$$ 0 0
$$653$$ −8.15796 −0.319246 −0.159623 0.987178i $$-0.551028\pi$$
−0.159623 + 0.987178i $$0.551028\pi$$
$$654$$ 0 0
$$655$$ −9.13159 −0.356801
$$656$$ 0 0
$$657$$ −15.8577 −0.618667
$$658$$ 0 0
$$659$$ 12.2222 0.476111 0.238055 0.971252i $$-0.423490\pi$$
0.238055 + 0.971252i $$0.423490\pi$$
$$660$$ 0 0
$$661$$ −22.7382 −0.884412 −0.442206 0.896914i $$-0.645804\pi$$
−0.442206 + 0.896914i $$0.645804\pi$$
$$662$$ 0 0
$$663$$ 9.66334 0.375293
$$664$$ 0 0
$$665$$ −2.29952 −0.0891715
$$666$$ 0 0
$$667$$ 3.16431 0.122522
$$668$$ 0 0
$$669$$ 12.9462 0.500531
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 18.0804 0.696949 0.348475 0.937318i $$-0.386700\pi$$
0.348475 + 0.937318i $$0.386700\pi$$
$$674$$ 0 0
$$675$$ 5.46097 0.210193
$$676$$ 0 0
$$677$$ −19.2431 −0.739572 −0.369786 0.929117i $$-0.620569\pi$$
−0.369786 + 0.929117i $$0.620569\pi$$
$$678$$ 0 0
$$679$$ 8.56790 0.328806
$$680$$ 0 0
$$681$$ 6.48469 0.248494
$$682$$ 0 0
$$683$$ −33.1235 −1.26743 −0.633717 0.773565i $$-0.718472\pi$$
−0.633717 + 0.773565i $$0.718472\pi$$
$$684$$ 0 0
$$685$$ 5.48597 0.209608
$$686$$ 0 0
$$687$$ −24.9419 −0.951591
$$688$$ 0 0
$$689$$ 8.57844 0.326813
$$690$$ 0 0
$$691$$ 43.3858 1.65047 0.825237 0.564787i $$-0.191042\pi$$
0.825237 + 0.564787i $$0.191042\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −17.7510 −0.673332
$$696$$ 0 0
$$697$$ 51.1035 1.93568
$$698$$ 0 0
$$699$$ 32.1911 1.21758
$$700$$ 0 0
$$701$$ 8.37727 0.316405 0.158203 0.987407i $$-0.449430\pi$$
0.158203 + 0.987407i $$0.449430\pi$$
$$702$$ 0 0
$$703$$ −14.3996 −0.543089
$$704$$ 0 0
$$705$$ −4.75255 −0.178992
$$706$$ 0 0
$$707$$ 18.5766 0.698644
$$708$$ 0 0
$$709$$ 0.475440 0.0178555 0.00892777 0.999960i $$-0.497158\pi$$
0.00892777 + 0.999960i $$0.497158\pi$$
$$710$$ 0 0
$$711$$ −10.2718 −0.385224
$$712$$ 0 0
$$713$$ 48.2566 1.80722
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −27.7680 −1.03701
$$718$$ 0 0
$$719$$ 1.66515 0.0620997 0.0310499 0.999518i $$-0.490115\pi$$
0.0310499 + 0.999518i $$0.490115\pi$$
$$720$$ 0 0
$$721$$ −6.82091 −0.254024
$$722$$ 0 0
$$723$$ 11.8589 0.441038
$$724$$ 0 0
$$725$$ 0.726479 0.0269808
$$726$$ 0 0
$$727$$ −14.5667 −0.540247 −0.270124 0.962826i $$-0.587065\pi$$
−0.270124 + 0.962826i $$0.587065\pi$$
$$728$$ 0 0
$$729$$ 22.3757 0.828730
$$730$$ 0 0
$$731$$ −3.91372 −0.144754
$$732$$ 0 0
$$733$$ 42.7581 1.57931 0.789653 0.613554i $$-0.210261\pi$$
0.789653 + 0.613554i $$0.210261\pi$$
$$734$$ 0 0
$$735$$ 4.32331 0.159468
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 45.0404 1.65684 0.828419 0.560109i $$-0.189241\pi$$
0.828419 + 0.560109i $$0.189241\pi$$
$$740$$ 0 0
$$741$$ −1.87483 −0.0688736
$$742$$ 0 0
$$743$$ −21.6833 −0.795482 −0.397741 0.917498i $$-0.630206\pi$$
−0.397741 + 0.917498i $$0.630206\pi$$
$$744$$ 0 0
$$745$$ 15.4005 0.564231
$$746$$ 0 0
$$747$$ −1.01513 −0.0371417
$$748$$ 0 0
$$749$$ 14.1382 0.516598
$$750$$ 0 0
$$751$$ 3.62118 0.132139 0.0660693 0.997815i $$-0.478954\pi$$
0.0660693 + 0.997815i $$0.478954\pi$$
$$752$$ 0 0
$$753$$ −21.2950 −0.776035
$$754$$ 0 0
$$755$$ −14.4892 −0.527316
$$756$$ 0 0
$$757$$ 32.3595 1.17612 0.588062 0.808816i $$-0.299891\pi$$
0.588062 + 0.808816i $$0.299891\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 26.4350 0.958268 0.479134 0.877742i $$-0.340951\pi$$
0.479134 + 0.877742i $$0.340951\pi$$
$$762$$ 0 0
$$763$$ −5.63428 −0.203975
$$764$$ 0 0
$$765$$ 10.1603 0.367348
$$766$$ 0 0
$$767$$ 1.77273 0.0640094
$$768$$ 0 0
$$769$$ 31.6062 1.13975 0.569874 0.821732i $$-0.306992\pi$$
0.569874 + 0.821732i $$0.306992\pi$$
$$770$$ 0 0
$$771$$ 33.3391 1.20068
$$772$$ 0 0
$$773$$ 28.3808 1.02079 0.510393 0.859941i $$-0.329500\pi$$
0.510393 + 0.859941i $$0.329500\pi$$
$$774$$ 0 0
$$775$$ 11.0790 0.397970
$$776$$ 0 0
$$777$$ −25.2444 −0.905639
$$778$$ 0 0
$$779$$ −9.91484 −0.355236
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 3.96729 0.141779
$$784$$ 0 0
$$785$$ −4.88487 −0.174349
$$786$$ 0 0
$$787$$ 29.1912 1.04056 0.520278 0.853997i $$-0.325828\pi$$
0.520278 + 0.853997i $$0.325828\pi$$
$$788$$ 0 0
$$789$$ 17.7467 0.631800
$$790$$ 0 0
$$791$$ 19.3284 0.687239
$$792$$ 0 0
$$793$$ 12.2434 0.434775
$$794$$ 0 0
$$795$$ −8.15525 −0.289237
$$796$$ 0 0
$$797$$ 4.47951 0.158672 0.0793362 0.996848i $$-0.474720\pi$$
0.0793362 + 0.996848i $$0.474720\pi$$
$$798$$ 0 0
$$799$$ −25.6795 −0.908475
$$800$$ 0 0
$$801$$ −14.8053 −0.523119
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −8.00509 −0.282142
$$806$$ 0 0
$$807$$ 9.52507 0.335298
$$808$$ 0 0
$$809$$ −41.6062 −1.46280 −0.731398 0.681951i $$-0.761132\pi$$
−0.731398 + 0.681951i $$0.761132\pi$$
$$810$$ 0 0
$$811$$ −18.0341 −0.633262 −0.316631 0.948549i $$-0.602552\pi$$
−0.316631 + 0.948549i $$0.602552\pi$$
$$812$$ 0 0
$$813$$ 22.4493 0.787333
$$814$$ 0 0
$$815$$ 23.0578 0.807680
$$816$$ 0 0
$$817$$ 0.759319 0.0265652
$$818$$ 0 0
$$819$$ 3.63522 0.127025
$$820$$ 0 0
$$821$$ −7.32591 −0.255676 −0.127838 0.991795i $$-0.540804\pi$$
−0.127838 + 0.991795i $$0.540804\pi$$
$$822$$ 0 0
$$823$$ −15.1762 −0.529008 −0.264504 0.964385i $$-0.585208\pi$$
−0.264504 + 0.964385i $$0.585208\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −52.0554 −1.81014 −0.905072 0.425257i $$-0.860184\pi$$
−0.905072 + 0.425257i $$0.860184\pi$$
$$828$$ 0 0
$$829$$ 29.1386 1.01203 0.506013 0.862526i $$-0.331119\pi$$
0.506013 + 0.862526i $$0.331119\pi$$
$$830$$ 0 0
$$831$$ −12.9625 −0.449664
$$832$$ 0 0
$$833$$ 23.3602 0.809381
$$834$$ 0 0
$$835$$ 3.24094 0.112157
$$836$$ 0 0
$$837$$ 60.5022 2.09126
$$838$$ 0 0
$$839$$ 42.4131 1.46426 0.732131 0.681163i $$-0.238526\pi$$
0.732131 + 0.681163i $$0.238526\pi$$
$$840$$ 0 0
$$841$$ −28.4722 −0.981801
$$842$$ 0 0
$$843$$ −28.7753 −0.991075
$$844$$ 0 0
$$845$$ −11.4238 −0.392991
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 10.8847 0.373560
$$850$$ 0 0
$$851$$ −50.1277 −1.71836
$$852$$ 0 0
$$853$$ −5.09345 −0.174396 −0.0871982 0.996191i $$-0.527791\pi$$
−0.0871982 + 0.996191i $$0.527791\pi$$
$$854$$ 0 0
$$855$$ −1.97125 −0.0674155
$$856$$ 0 0
$$857$$ −27.4535 −0.937795 −0.468897 0.883253i $$-0.655349\pi$$
−0.468897 + 0.883253i $$0.655349\pi$$
$$858$$ 0 0
$$859$$ −27.9580 −0.953914 −0.476957 0.878927i $$-0.658260\pi$$
−0.476957 + 0.878927i $$0.658260\pi$$
$$860$$ 0 0
$$861$$ −17.3821 −0.592380
$$862$$ 0 0
$$863$$ 1.45954 0.0496834 0.0248417 0.999691i $$-0.492092\pi$$
0.0248417 + 0.999691i $$0.492092\pi$$
$$864$$ 0 0
$$865$$ −17.6970 −0.601718
$$866$$ 0 0
$$867$$ −29.3482 −0.996717
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −15.8062 −0.535572
$$872$$ 0 0
$$873$$ 7.34481 0.248584
$$874$$ 0 0
$$875$$ −1.83785 −0.0621308
$$876$$ 0 0
$$877$$ −3.94038 −0.133057 −0.0665287 0.997785i $$-0.521192\pi$$
−0.0665287 + 0.997785i $$0.521192\pi$$
$$878$$ 0 0
$$879$$ 31.3048 1.05588
$$880$$ 0 0
$$881$$ −44.6771 −1.50521 −0.752604 0.658473i $$-0.771203\pi$$
−0.752604 + 0.658473i $$0.771203\pi$$
$$882$$ 0 0
$$883$$ −47.5197 −1.59916 −0.799582 0.600557i $$-0.794946\pi$$
−0.799582 + 0.600557i $$0.794946\pi$$
$$884$$ 0 0
$$885$$ −1.68527 −0.0566498
$$886$$ 0 0
$$887$$ −45.7025 −1.53454 −0.767270 0.641325i $$-0.778385\pi$$
−0.767270 + 0.641325i $$0.778385\pi$$
$$888$$ 0 0
$$889$$ 10.9900 0.368591
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 4.98220 0.166723
$$894$$ 0 0
$$895$$ −17.6078 −0.588565
$$896$$ 0 0
$$897$$ −6.52666 −0.217919
$$898$$ 0 0
$$899$$ 8.04867 0.268438
$$900$$ 0 0
$$901$$ −44.0653 −1.46803
$$902$$ 0 0
$$903$$ 1.33119 0.0442993
$$904$$ 0 0
$$905$$ −4.79712 −0.159462
$$906$$ 0 0
$$907$$ 36.5095 1.21228 0.606139 0.795359i $$-0.292718\pi$$
0.606139 + 0.795359i $$0.292718\pi$$
$$908$$ 0 0
$$909$$ 15.9247 0.528189
$$910$$ 0 0
$$911$$ 22.1432 0.733637 0.366819 0.930293i $$-0.380447\pi$$
0.366819 + 0.930293i $$0.380447\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ −11.6394 −0.384786
$$916$$ 0 0
$$917$$ 16.7825 0.554208
$$918$$ 0 0
$$919$$ −54.4511 −1.79618 −0.898088 0.439816i $$-0.855044\pi$$
−0.898088 + 0.439816i $$0.855044\pi$$
$$920$$ 0 0
$$921$$ 9.73958 0.320930
$$922$$ 0 0
$$923$$ 8.53293 0.280865
$$924$$ 0 0
$$925$$ −11.5086 −0.378401
$$926$$ 0 0
$$927$$ −5.84720 −0.192047
$$928$$ 0 0
$$929$$ −0.727990 −0.0238846 −0.0119423 0.999929i $$-0.503801\pi$$
−0.0119423 + 0.999929i $$0.503801\pi$$
$$930$$ 0 0
$$931$$ −4.53221 −0.148537
$$932$$ 0 0
$$933$$ −16.1047 −0.527243
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −54.3101 −1.77423 −0.887116 0.461546i $$-0.847295\pi$$
−0.887116 + 0.461546i $$0.847295\pi$$
$$938$$ 0 0
$$939$$ −9.35511 −0.305292
$$940$$ 0 0
$$941$$ 19.3033 0.629270 0.314635 0.949213i $$-0.398118\pi$$
0.314635 + 0.949213i $$0.398118\pi$$
$$942$$ 0 0
$$943$$ −34.5155 −1.12398
$$944$$ 0 0
$$945$$ −10.0365 −0.326486
$$946$$ 0 0
$$947$$ −8.04409 −0.261398 −0.130699 0.991422i $$-0.541722\pi$$
−0.130699 + 0.991422i $$0.541722\pi$$
$$948$$ 0 0
$$949$$ 12.6365 0.410198
$$950$$ 0 0
$$951$$ −7.55274 −0.244914
$$952$$ 0 0
$$953$$ −39.1448 −1.26803 −0.634013 0.773323i $$-0.718593\pi$$
−0.634013 + 0.773323i $$0.718593\pi$$
$$954$$ 0 0
$$955$$ 11.6292 0.376312
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −10.0824 −0.325578
$$960$$ 0 0
$$961$$ 91.7444 2.95950
$$962$$ 0 0
$$963$$ 12.1199 0.390559
$$964$$ 0 0
$$965$$ −4.05323 −0.130478
$$966$$ 0 0
$$967$$ 38.4906 1.23777 0.618887 0.785480i $$-0.287584\pi$$
0.618887 + 0.785480i $$0.287584\pi$$
$$968$$ 0 0
$$969$$ 9.63053 0.309377
$$970$$ 0 0
$$971$$ 53.8570 1.72835 0.864176 0.503189i $$-0.167840\pi$$
0.864176 + 0.503189i $$0.167840\pi$$
$$972$$ 0 0
$$973$$ 32.6237 1.04587
$$974$$ 0 0
$$975$$ −1.49843 −0.0479881
$$976$$ 0 0
$$977$$ −10.6010 −0.339156 −0.169578 0.985517i $$-0.554240\pi$$
−0.169578 + 0.985517i $$0.554240\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −4.82997 −0.154209
$$982$$ 0 0
$$983$$ 17.3284 0.552691 0.276346 0.961058i $$-0.410877\pi$$
0.276346 + 0.961058i $$0.410877\pi$$
$$984$$ 0 0
$$985$$ 17.4066 0.554620
$$986$$ 0 0
$$987$$ 8.73450 0.278022
$$988$$ 0 0
$$989$$ 2.64334 0.0840534
$$990$$ 0 0
$$991$$ 1.68386 0.0534894 0.0267447 0.999642i $$-0.491486\pi$$
0.0267447 + 0.999642i $$0.491486\pi$$
$$992$$ 0 0
$$993$$ −3.48978 −0.110745
$$994$$ 0 0
$$995$$ −22.0409 −0.698742
$$996$$ 0 0
$$997$$ 16.7562 0.530674 0.265337 0.964156i $$-0.414517\pi$$
0.265337 + 0.964156i $$0.414517\pi$$
$$998$$ 0 0
$$999$$ −62.8482 −1.98843
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9680.2.a.ct.1.1 4
4.3 odd 2 4840.2.a.z.1.4 4
11.7 odd 10 880.2.bo.d.401.2 8
11.8 odd 10 880.2.bo.d.801.2 8
11.10 odd 2 9680.2.a.cu.1.1 4
44.7 even 10 440.2.y.a.401.1 yes 8
44.19 even 10 440.2.y.a.361.1 8
44.43 even 2 4840.2.a.y.1.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.a.361.1 8 44.19 even 10
440.2.y.a.401.1 yes 8 44.7 even 10
880.2.bo.d.401.2 8 11.7 odd 10
880.2.bo.d.801.2 8 11.8 odd 10
4840.2.a.y.1.4 4 44.43 even 2
4840.2.a.z.1.4 4 4.3 odd 2
9680.2.a.ct.1.1 4 1.1 even 1 trivial
9680.2.a.cu.1.1 4 11.10 odd 2